Colloid Polym Sci DOI 10.1007/s00396-013-2983-0

ORIGINAL CONTRIBUTION

Thermodynamic studies on thin liquid films. V. General formulation for spherical films Hidemi Iyota

Received: 20 February 2013 / Revised: 11 April 2013 / Accepted: 5 May 2013 # Springer-Verlag Berlin Heidelberg 2013

Abstract Thermodynamic equations were derived for adsorption at interfaces of spherical films and interaction between interfaces in a film. The dependence of film tension on capillary pressure between external and filmforming phases gave thermodynamic film thickness and the one on pressure difference across the film the distance between the surface of tension for the film and the innermost dividing surface. Curvature dependence of film tension was numerically evaluated by using one of the derived equations. Keywords Spherical films . Thermodynamics and quasithermodynamics . Adsorption at film interfaces . Thermodynamic film thickness . Curvature dependence of film tension . Interaction between interfaces

Introduction Curved thin films are formed when two bubbles or emulsion droplets of different size contact. Curved films of vesicles and floating bubbles play an important role in their stability and fusion [1]. Physicochemical properties of curved films are also related to the gas permeation through a foam film caused by pressure difference across the film [2]. Clarifying the structure and properties of curved thin films and adsorbed monolayers at the film interfaces is crucial not only in the fundamental studies, but in the studies of biological membrane and gas exchange through thin liquid films in lung cavities.

H. Iyota (*) Kagoshima Prefectural College, 1-52-1, Shimo-ishiki, Kagoshima 890-0005, Japan e-mail: [email protected]

It is essential to get thermodynamic quantities of thin liquid films to elucidate the structure and properties of the films [3–6]. For spherical films as well as plane-parallel films, thermodynamic equations have been developed by choosing one or two spherical dividing surfaces and taking at least one chemical potential of solvent as thermodynamic variable [7]. Dependence of film tension on the chemical potential of solvent leads to the adsorption of solvent. On the other hand, pressure in the bulk phases separated by the film was taken instead of the chemical potential of solvent as a variable in the thermodynamic description with four dividing planes in part I of this series [4]. The volume change associated with adsorption of solutes at the film surfaces can be obtained from the dependence of film tension on the pressure. The volume change is superior to the adsorption of solvent in understanding the structure and properties of the film, like adsorption of solutes at a bulk interface [8, 9]. Furthermore, thermodynamic equations including the chemical potential of solvent as variable in the previous studies [3, 7] are approximate ones when they are applied to film tension to obtain the adsorption of solutes because the film tension is usually measured in the presence of air under atmospheric pressure [5]. In the present study, thermodynamic formulation developed for plane-parallel films in part I [4] is extended so as to be applicable to spherical films. The system of a spherical film has one more degree of freedom compared with the system of the corresponding plane-parallel film and curvature of the spherical film or pressure difference across the film is taken as a variable in thermodynamic description of the film. Equations are derived for adsorption of solutes at interfaces of a spherical film and interaction between interfaces in the film. The dependence of film tension on the pressure difference across the film or the radius of the film is correlated to the distance between the surface of tension for the film and the innermost dividing surface.

Colloid Polym Sci

Fundamental equations Let us consider a system consisting of inner phase A and outer one B separated by a spherical thin film formed from phase L, enclosed in a spherical cone of solid angle ω (Fig. 1). Components a, b, and l are the solvents in the phases A, B, and L, respectively, and 1,…, c are the solutes, which are taken to be nonionic for simplicity. The effects of gravity and line tension are assumed to be negligible. At equilibrium, temperature T and chemical potentials μa, μb, μl, μ1,…, μc of the components are uniform throughout the system. On the other hand, pressure pA in the phase A is different from that p in the phase B. The Gibbs–Duhem equation for the system can be expressed by 



−SdT þ V A dpA þ V B dp−σdg f −na dμa c X −nb dμb −nl dμl − ni dμi ¼ 0;

ð1Þ

i¼1

where S denotes the entropy of the system, V A* and V B* the volumes of portions A* and B* of the system separated by the surface of tension (of radius r) for film, σ (=ω r2) the film area, g f the film tension, and na, nb, nl, n1,…, nc the numbers of moles of the components. The Gibbs–Duhem equations for the phase A is expressed by A A −sA dT þ dpA −cA a dμa −cb dμb −cl dμl −

c X i¼1

Fig. 1 Schematic illustrations of: a a spherical film formed from the phase L between the phases A and B and b regions in the spherical film separated by the surfaces of tension and the dividing surfaces, concentric with the film. Dashed line: surface of tension; straight line: dividing surface

cA i dμi ¼ 0;

ð2Þ

where sA and ciA denote the entropy and the number of moles of component i per unit volume in the phase A. The corresponding equations for the phases B and L are the same as Eqs. 3 and 4 for plane-parallel films in part I [4] expressed by −sB dT þ dp−cBa dμa −cBb dμ b −cBl dμl −

c X

cBi dμi ¼ 0;

ð3Þ

cLi dμi ¼ 0:

ð4Þ

i¼1

−sL dT þ dpL −cLa dμa −cLb dμb −cLl dμl −

c X i¼1

Subtracting Eqs. 2–4 multiplied by the volumes V A, V B, and V L of the phases A, B, and L, respectively, from Eq. 1, and employing the same convention (film densities of solvents are zero) as that used for plane-parallel films in part I, we have c X     dg f ¼ −s f dT þ vAf þ vLA dpA þ vBf þ vLB dp−vL dpL − Γ if dμi i¼1

ð5Þ

c X   ¼ −s f dT þ v f dp þ vL dpBL þ vAf þ vLA dpAB − Γ if dμi ; i¼1

ð6Þ

Colloid Polym Sci

where film density Γif, excess entropy sf and volume vf per unit film area, and volume vL of the phase L per unit film area are defined by the same equations as those for plane-parallel films in part I:   B B L L Γ if ¼ nif =σ ¼ ni −V A cA i −V ci −V ci =σ;

ði ¼ 1; …; cÞ;

ð7Þ

  B B L L y f ¼ Y f =σ − Y −V A cA i −V ci −V ci =σ;

ðy ¼ s; vÞ;

ð8Þ

and vL ¼ V L =σ;

ð9Þ

where nif is the excess quantity of component i and Y f the corresponding thermodynamic quantities. vAf and vBf are the excess volume per unit film area ascribable to the film interfaces against the phases A and B, respectively, and vL is divided into vAL and vBL by the surface of tension for the film, as shown in Fig. 1b. pBL is the pressure difference between the phases B and L and pAB the one across the film, pBL ¼ p−pL

ð10Þ

chosen so that the film densities of solvents are simultaneously zero (Fig. 2): Γ af ¼ Γ bf ¼ Γ lf ¼ 0:

ð12Þ

Equation 5 is the fundamental equation for the spherical film. Equations 5 and 6 show the total differential of g f as a function of c+4 independent variables, which is consistent with the phase rule for a capillary system with a spherical film [10]. Let us consider the adsorption of solutes at film interfaces from the phase L and take T, p, pBL, pAB, and the molalities of solutes m1,…, mc as experimental variables of the system. The total differential of the chemical potential of component i can be expressed by       dμi ¼ − si þ D−1 Dbs μia þ Das μib dT þ vi þ D−1 Dbv μia þ Dav μib dp       − vi −D−1 DbL μia þ DaL μib dpBL þ D−1 DbA μia þ DaA μib dpAB  i c h X þ μij −D−1 Dbj μia þ Daj μib dm j ; ði ¼ a; b; l; 1; ⋯; cÞ j¼1

ð13Þ where DAa and DAb are the determinants    A   A DA  D  DA A ; D b ¼  D A b ; DaA ¼  Ba A  0 D 0  DB  a

ð14Þ

b

and pAB ¼ pA −p:

ð11Þ

DA A being A A A A A A DA A ¼ ma va þ mb vb þ ml vl þ

pAB is a variable characteristic of the spherical film and for equilibrium plane-parallel films pBL is the disjoining pressure. The locations of spherical dividing surfaces (concentric with the film) which decide VA, V B, and V L were

c X

A; mA i vi

ð15Þ

i¼1

DK j (j=a,

and b; K=A, B) (and the other determinants are defined by Eqs. II.10 and II.16 in part I (see Appendix II in ref. 4)).

Fig. 2 Concentration profiles of the solvents a, b, l and solute i in the spherical film and the locations of four dividing surfaces and the film density of solute i relative to the solvents

Colloid Polym Sci

Substitution of Eq. 13 into Eq. 6 leads to ( ) h  i c X dT dg f ¼ − s f − Γ fj s j þ D−1 Dbs μja þ Das μjb j¼1 ( ) h  i c X þ vf − dp Γ fj v j þ D−1 Dbv μja þ Dav μjb j¼1 ( ) h  i c X þ vL þ dpBL Γ fj v j  D−1 DbL μja þ DaL μjb j¼1 " #   c X f L −1 f b a þ vA þ vA −D Γ j DA μja þ DA μjb dpAB j¼1 ( ) c h  i c X X f −1 b a − Γ j μji −D Di μja þ Di μjb dmi : i¼1

j¼1

ð16Þ It may be convenient to take the film radius r as an experimental variable instead of pAB and then Eq. 16 can be rewritten by (

" 1−2

vAf

# )  . r dgf ¼ þ Γ þ j¼1 ( ) h  i c X − sf − Γ fj s j þ D−1 Dbs μja þ Das μjb dT ( j¼1 ) h  i c X Γ fj v j þ D−1 Dbv μja þ Dav μjb þ vf− dp j¼1 ( ) h  i c X Γ fj v j −D−1 DbL μja þ DaL μjb dpBL þ vL þ j¼1 (" ) #   . c X −2 vAf þ vLA −D−1 Γ fj DbA μja þ DaA μjb g f r2 dr j¼1 ( ) h  i c c X X f Γ j μji −D−1 Dbi μja þ Dai μjb − dmi ; vLA −D−1

i¼1

c X

f j



DbA μja

DaA μjb

j¼1

ð17Þ

where we used an analogue of the Laplace equation for the film

pAB ¼ 2g f =r:

ð18Þ

Equations 16 and 17 show that sf, vf, vL, vAf +vAL,Γ 1f,IIIΓ fc can be obtained from g f as a function of T, p, pBL, pAB (or r), and m1,…, mc. Thermodynamic equations for adsorption at film interfaces from the other phases can be derived as well. Equation 1 can be rewritten in terms of film interfacial tension g AF and g BF against the phases A and B [11, 12], respectively, as

  −SdT þ V dp þ V A þ V AF;A dpAB −σvh dpBL −σpBL dvh c X ; −σA dg AF −σB dg BF −na dμa −nb dμb −nl dμl − ni dμi ¼ 0

ð19Þ

i¼1

where σA(=ω rA2) and σB(=ω rB2) are the area of the surfaces of tension for the film interfaces against the phases A and B, respectively, rA and rB being the corresponding radii of the surfaces of tension (Appendix 1). The surfaces of tension divide VAF into VAF,A and VAF,L and VBF into VBF,B and VBF,L as shown in Fig. 1b. vh, the volume Vh between the surfaces of tension for the film surfaces divided by σ, is related to the distances ht and htA between those interfaces,

     vh ¼ V hh =σ ¼ V L þ V AF;L þ V BF;L =σ ¼ 1=3r2 r3B −r3A i    A 2  A  2 ¼ ht 1−2 hA =r þ ð h =r Þ þ h =r − h =r ð h =r Þ þ ð 1=3 Þ ð h =r Þ ; t t t t t t

ð20Þ

ht and hAt being

ð22Þ

hA t ¼ r − r A:

ht ¼ rB −rA

ð21Þ

and

vh is similar to ht for a film of which the radius is much larger than its thickness. We can obtain the equation corresponding to Eq. 6 by using Eq. 19 instead of Eq. 1:

        r−2 r2A dg AF þ r2B dg BF ¼ −r−2 r2A sAF þ r2B sBF dT þ r−2 r2A vAF þ r2B vBF dp− vh −vL dpBL   c X þ r−2 r2A vAF;A d pAB − pBL d vh − r−2 r2A Γ iF;A þ r2B Γ iF;B dμi ; i¼1

ð23Þ

Colloid Polym Sci

where interfacial density ΓiF,K of component i of film interface against phase K and the corresponding thermodynamic quantity yKF are defined by    K K L L Γ iF;K ¼ niF;K =σK ¼ nK i −V cl −V K ci =σK ; ði ¼ 1; ⋯; c; K ¼ A; BÞ;

ð24Þ

and

   yKF ¼ Y KF =σK ¼ Y K −V K yK −V LK yL =σK ; ðy ¼ s; v; K ¼ A; BÞ;

ð25Þ niK* and YK* being the number of moles of component i of portion K* and the corresponding thermodynamic quantity, respectively, and vKF,K′ is

0

0

vKF;K ¼ V KF;K =σK ;

ðK ¼ A; B; K0 ¼ K; LÞ:

ð26Þ

By definition,   Γ if ¼ r−2 r2A Γ iF;A þ r2B Γ iF;B

ð27Þ

and   y f ¼ r−2 r2A yAF þ r2B yBF :

ð28Þ

Substituting Eq. 13 into Eq. 23 and expressing vh as a function of T, p, pBL, pAB, and m1,…, mc, we have

  r−2 r2A dg AF þ r2B dg BF ¼ c  h  i   X − r−2 r2A sAF þ r2B sBF − r−2 r2A Γ F;A þ r2B Γ F;B s j þ D−1 Dbs μja þ Das μjb j j

f

þp

BL

( þ −p



.

∂vh ∂T

p; pBL ; pAB ; m

gdT

 h  i c   X r−2 r2A vAF þ r2B vBF − r−2 r2A Γ F;A þ r2B Γ F;B v j þ D−1 Dbv μja þ Dav μjb j j

BL



.  ∂vh ∂p

( þ

j¼1



vL þ "

c X

j¼1 T; pBL ; pAB ; m

gdp

 h  i r−2 r2A Γ F;A þ r2B Γ F;B v j −D−1 DbL μja þ DaL μjb j j

j¼1

h  i . − ∂ pBL vh ∂pBL

T ; p; pAB ; m

gdpBL

c     .  X 2 F;B b a BL AB þ r−2 r2A vAF;A −D−1 r−2 r2A Γ F;A þ r Γ μ þ D μ ∂v dpAB D −p ∂p h j B j A ja A jb T ; p; pBL ; m j¼1 (  h  i  .  c c X X −2 2 F;A − r rA Γ j þ r2B Γ F;B gdmi : μji −D−1 Dbi μja þ Dai μjb þ pBL ∂vh ∂mi j BL AB i¼1

T ; p; p ; p

j¼1

where the subscript m of partial derivatives denotes that m1,…, mc are kept constant and the subscript mi all the molalities except mi are kept constant. It is noticed from Eq. 29 that the thermodynamic quantities for each film interface cannot be obtained

        d γ AF þ γ BF ¼ − sAF þ sBF dT þ vAF þ vBF dp− t−vL dΠ−Πdt c   X Γ iF;A þ Γ iF;B dμi ; − i¼1

ð29Þ

; mi

separately even for spherical foam and emulsion films, unlike symmetric plane-parallel films [4]. When r, rA, and rB approach infinity at constant ht and htA, Eq. 23 reduces to Eq. 15 in part I for plane-parallel films given by

ð30Þ

Colloid Polym Sci

in which pBL is the disjoining pressure Π and ht is denoted by C.

Thermodynamic film thickness and change in thermodynamic quantity associated with adsorption

quantities [13, 14]. Let the distance from the origin of the system be r′ and the radii of the spherical dividing surfaces rAF, rAL, rBL, and rBF as shown in Fig. 1b. Film region is shown by IA
Quasi-thermodynamics correlates local thermodynamic quantities of inhomogeneous regions to their excess thermodynamic Z IB  0 2 0  0 ABL 0  0 2 0 0 −2 ci ðr0 Þ−cABL ð r Þ ð r Þ dr ¼ r ci ðr Þ−ci ðr Þ ðr Þ dr i IA 0  I;B I;L ¼ Γ Ii − Γ I;A ; ði ¼ a; b; l; 1; …; cÞ; i þ Γi þ Γi

Γ if ¼ r−2

Z

∞

where ciABL(r′) is ðr 0 Þ cABL i

I;B I;L Γ Ii ¼ Γ I;A i þ Γi þ Γi ; 0

¼ ½1−Aðr −rAF ÞcA i 0 B þ Aðr −rBF Þci ;

0

þ ½Aðr −rAL Þ−Aðr

0

−rBL ÞcLi ð32Þ

A(r′) being the step function Aðr0 Þ ¼ 1; ðr0 ≥ 0Þ; ¼ 0; ðr0 < 0˙:Þ

ð33Þ

Here Γi is the film density of component i inherent in the film, and ΓiI,A, ΓiI,B, and ΓiI,L are the ones inherent in the portions of the film region separated by the dividing surfaces, expressed by Z IB 2 Γ Ii ¼ r−2 ci ðr0 Þðr0 Þ dr0 ; ð34Þ IA

Z

¼

r−2 cA i

Z

Γ I;B i

¼

r−2 cBi

Z

−2 L Γ I;L i ¼ r ci

rAF

2

ðr0 Þ dr0 ; 0

ðr Þ dr ; 2

ðr0 Þ dr0 ;

rAL

respectively. From Eq. 12, we have

Z r−2

ð34bÞ

rBF rBL

IA

i¼a

where yABL ðr0 Þ ¼ ½1−Aðr0 −rAF ÞyA þ ½Aðr0 −rAL Þ−Aðr0 −rBL ÞyL þAðr0 −rBF ÞyB; ð37Þ and we used the relation between thermodynamic quantity y and partial molar quantity yi for the bulk phases

y¼ 0 2

ð35Þ

On the other hand, the excess thermodynamic quantity can be expressed by Z ∞  0  2 y f ¼ r−2 yðr Þ − yABL ðr0 Þ ðr0 Þ dr0 0 Z c IB  X  2 ¼ r−2 ci ðr0 Þyi ðr0 Þ−cABL ðr0 ÞyABL ðr0 Þ ðr0 Þ dr0 ; i i

ð34aÞ

IA IB

ði ¼ a; b; l Þ:

ð36Þ

I

Γ I;A i

ð31Þ

ð34cÞ

C X

ci yi;

ð38Þ

i¼a

and the corresponding one between y(r′) and yi(r′) for the film region c X yð r 0 Þ ¼ ci ðr0 Þyi ðr0 Þ: ð39Þ i¼a

The integral in Eq. 36 can be reduced to

Z IB  0 0 2 0 ABL 0 −2 ci ðr0 Þyi ðr0 Þ−cABL ð r Þy ð r Þ ð r Þdr ¼ r ci ðr0 Þyi ðr0 Þðr0 Þ dr0 i i IA I A

Z rAF Z IB Z rBL −2 A A 0 2 0 B B 0 2 0 L 0 2 0 − r ci yi ðr Þ dr þ ci yi ðr Þ dr þci yi ðr Þ dr IA rBF rAL   I;B B I;L A ¼ Γ Ii yiI − Γ I;A i yi þ Γ i yi þ Γ i yi IB 

ð40Þ

Colloid Polym Sci

¼ Γ Ii Δyi ;

ði ¼ a; b; l Þ;

ð40aÞ

where the mean partial molar quantities yIi and yif are defined by

Z I yi ¼

IB

ci ðr0 Þyi ðr0 Þðr0 Þ dr0 2

.Z

IA f

¼ Γ if yi ; ði ¼ 1; …; cÞ;

Z f yi ¼

IB  IA

ð40bÞ

Z  0 2 0. 0 ABL 0 ci ðr0 Þyi ðr0 Þ−cABL ð r Þy ð r Þ ð r Þ dr i i

IB  IA

I

I;B B I;L A Δyi ¼ yi − Γ I;A i yi þ Γ i yi þ Γ i yi

.

Γ Ii :

c X

f

Γ if yi :

ð44Þ

We hence define change Δyf in thermodynamic quantity associated with the adsorption of solutes at film interfaces from the phase L, by c X i¼1

þΓ Il Δyl

þ

ð45Þ

Γ if yi ¼ Γ Ia Δya þ Γ Ib Δyb c X

Γ if



f yi −yi

and

. cLl ; vL ¼ Γ I;L l

¼ − 1−2 −1

þD

c X j¼1

þ Γ

vLA −D−1 

f j

ð46Þ

h f ¼ vL þ

c X

. c X Γ if vi ¼ Γ I;L Γ if vi ; cLl þ l

i¼1

ð47Þ

i¼1

where the first term in the right-hand side is a measure of the thickness of film core equivalent to the phase L and the second one the total thickness of adsorbed films at film interfaces reduced to the partial molar volumes of solutes in the phase L. It is noticed that Eqs. 45 and 47 are identical with the corresponding ones for plane-parallel films in part I. Substituting Eqs. 45 and 47 into Eqs. 16, 17, and 29 we have,   Δs f ¼ − ∂γ f =∂Τ p;pBL ;pAB ;m



þ D−1

c X

  Γ fj Dbs μja þ Das μjb

ð48Þ

j¼1

" vAf

ð42Þ

From Eq. 34c,

i¼1

(

ð41Þ

IA

thermodynamic film thickness hf can then be defined by

i¼1

Δy f ¼ y f −

2

ð43Þ

ΓiI and Γif, and yIi and yif should be distinguished from each other because ΓiI and yIi are the quantities inherent in the film and Γif and yif are the excess ones defined with respect to the dividing surfaces. Substituting Eqs. 40a and 40b into Eq. 36 yields y f ¼ Γ Ia Δya þ Γ Ib Δyb þ Γ Il Δyl þ

ci ðr0 Þðr0 Þ dr0

 2 ci ðr0 Þ−cABL ðr0 Þ ðr0 Þ dr0 ; i

respectively, and change Δyi in the partial molar quantity of solvent i associated with adsorption is written by 

IB

c X

Γ

f j



j¼1

Dbs μja þ Das μjb

DbA μja 

þ

DaA μjb

# )  .  .  r ∂γ f ∂T

p;pBL ;r;m

ð48aÞ

Colloid Polym Sci

h  i    ¼ −r−2 r2A ∂γ AF =∂T p; pBL ; pAB ; m þ r2B ∂γ BF =∂T p; pBL ; pAB ; m

c   X   Δv f ¼ ∂γ f =∂p T; pBL ; pAB ; m þ D−1 Γ fj Dbv μja þ Dav μjb

−pBL ð∂vh =∂T Þp; pBL ; pAB ; þD−1

c X

   r−2 r2A Γ F;A þ r2B Γ F;B Dbs μja þ Das μjb ; j j

j¼1

ð49Þ

j¼1

ð48bÞ (

" 1−2 vAf þ vLA −D−1

¼

þD

−1

c X

Γ

f j



c X

Γ fj



j¼1

Dbv μja

þ

# )  .  .  DbA μja þ DaA μjb r ∂γ f ∂p

T ; pBL ; r; m



Dav μjb

ð49aÞ

j¼1



.   .  2 F þ r ∂γ ∂p ¼ r−2 r2A ∂γ AF ∂p B B BL AB T ; pBL ; pAB ; m  .  T; p ; p ; m BL þp ∂vh ∂p T; pBL ; pAB ; m c    X þD−1 r−2 r2A Γ F;A þ r2B Γ F;B Dbv μja þ Dav μjb ; j j

c   X   h f ¼ ∂γ f =∂pBL T ; p; pAB ; m þ D−1 Γ fj DbL μja þ DaL μjb j¼1

ð50Þ

j¼1

ð49bÞ ( ¼

" vAf

1−2 þD−1

c X

þ

vLA −D−1

c X

Γ

f j



DbA μja

j¼1



Γ fj DbL μja þ DaL μjb

þ

DaA μjb

# )  .  .  r ∂γ f ∂pBL



T;p;r;m

ð50aÞ

j¼1



.   .  2 F BL ¼ r−2 r2A ∂γ AF ∂pBL þ r ∂γ ∂p B B AB T ; p; pAB ; m h  i T ; p; p ; m .  þ ∂ pBL vh ∂pBL T ; p; pAB ; m    c X −1 −2 2 F;A þD r rA Γ j þ r2B Γ F;B DbL μja þ DaL μjb ; j

 .  vAf þ vLA ¼ ∂γ f ∂pAB

T ; p; pBL ; m

þ D−1

j¼1

c X

  Γ fj DbA μja þ DaA μ jb

ð51Þ

j¼1

ð50bÞ .   .   ¼ 1 2 r2 ∂γ f ∂r

r−2 r2A vAf ; A

T; p; pBL ; m

.  .  r ∂γ f ∂r

 .  ¼ r r2A ∂γ AF ∂pAB

T ; p; pBL ; m

−2

T; p; pBL ; m

þ

r2B



  c X −γ f þ D−1 Γ fj DbA μja þ DaA μjb ;

∂γ BF

.

∂p

AB



 T ; p; pBL ; m

 .  þpBL ∂vh ∂pAB

T ; p; pBL ; m

þD−1

c X j¼1

ð51aÞ

j¼1

   r−2 r2A Γ F;A þ r2B Γ F;B DbA μja þ DaA μjb ; j j

ð51bÞ

Colloid Polym Sci c X

 .  Γ fj μji ¼ − ∂γ f ∂mi

T ;p;pBL ;pAB ;mi

j¼1

(

"

¼ − 1−2

vAf

þ

vLA −D−1

c X

Γ

f j



þ D−1

c X

  Γ fj Dbi μja þ Dai μjb

DbA μja

þ

DaA μjb

# )  .  .  r ∂γ f ∂mi

T ;p;pBL ;r;mi

j¼1

þD−1

c X

ð52Þ

j¼1

ð52aÞ

  Γ fj Dbi μja þ Dai μjb

j¼1

 .  ¼ −r r2A ∂γ AF ∂mi −2

T ; p; pBL ; pAB ; mi

 .  −pBL ∂vh ∂mi þD−1

c X j¼1

þ

r2B



∂γ BF

.

∂mi



 T ; p; pBL ; pAB ; mi

ð52bÞ

T ;p;pBL ;pAB ;mi

   2 F;B b a r−2 r2A Γ F;A þ r Γ μ þ D μ D j B j i ja i jb ;

ði ¼ 1; ::::; cÞ;

where Δyf is related to the corresponding quantity ΔyKF for each film interface,   ð53Þ Δy f ¼ r−2 r2A ΔyAF þ r2B ΔyBF ; ðy ¼ s; vÞ; ΔyKF ¼ yKF −

c X

Γ iF;K yi ;

ðK ¼ A; BÞ;

ð54Þ

i¼1

and we used the relation vAf þ vLA ¼ vAf ;A þ vhAt ; vAf,A

ð55Þ

entropy and volume changes associated with the adsorption of solutes at film interfaces, the thermodynamic film thickness, the volume per unit film area between the surface of tension for film and the innermost dividing surface, and the film densities of solute components, by applying Eqs. 48–52b to the film tension or the film interfacial tension and film thickness as a function of the experimental variables. The corresponding energy change Δuf and enthalpy change Δhf can also be evaluated. Combination of Eq. 45 and the following equations

and vhAt being defined by 

0

0

0

vKf ;K ¼ V KF;K =σ ¼ r−2 r2K vKF;K ; ðK ¼ A; B; K0 ¼ K; LÞ; ð56Þ

     vhAt ¼ V AF;L þ V LA =σ ¼ 1=3r2 r3 −r3A  A   A ¼ hA t 1− h t =r þ ð1=3Þ h t =r

2 i

ð58Þ

    ¼ U þ pA V A þ V AF;A þ p V B þ V BF;B

and

h



H ¼ U þ pA V A þ pV B −γ f σ

ð57Þ ;

respectively. vhAt is similar to h At for the film whose radius is very large compared with its thickness. We can thus obtain the

þ pL V h −γAF σA −γBF σB ;

ð58aÞ

and G ¼ H−TS;

ð59Þ

Colloid Polym Sci

U, H, and G being the energy, enthalpy, and Gibbs free energy of the system respectively, leads to

of radius r for the film and the innermost dividing surface of rAF (Fig. 1b),

Δg f ¼ Δh f −TΔs f ;

     3 vAf þ vLA ¼ VhAF þ V LA =σ ¼ 1=3r2 r3 −r i AF ¼ hA 1−ðhA =rÞ þ ð1=3ÞðhA =rÞ2 ;

ð60Þ

and   Δu f ¼ Δh f −pΔv f −pBL h f −pAB vAf ;A þ vhAt þ γ f



f

BL

f

where hA ¼ r − rAF :



ð67Þ

−vh −pAB vAf ;A

¼ Δh −pΔv −p h   þ r−2 r2A γAF þ r2B γBF f

ð61Þ

ð66Þ

ð61aÞ

Since Δ gf =0 by definition,   Δu f ¼ TΔs f −pΔv f −pBL h f −pAB vAf ;A þ vhAt þ γ f

  ¼ TΔs f −pΔv f −pBL h f −vh −pAB vAf ;A   þ r−2 r2A γAF þ r2B γBF

ð62Þ

ð62aÞ

and Δh f ¼ TΔs f :

ð63Þ

Curvature dependence of film tension

Dependence of hA on r is required to solve Eq. 65. When hA is independent of r and equal to hA∞, hA of a plane-parallel film, film tension can be calculated as a function of r by solving Eq. 65 (Appendix 2). Figure 3 shows numerical values of film tension in the form of the ratio of +f to +∞f , +f at r=∞, as a function of 1/r for typical values of the thickness of common black film (10 nm) and Newton black film (4 nm). It is noticed that film tension increases with decreasing film radius and film tension of the thicker film is larger than that of the thinner film at a given film radius and film thickness highly affects the curvature dependence. Film tension and interfacial tension are similarly treated in mechanical and thermodynamic descriptions of inhomogeneous regions, thin liquid films, and interfaces [15]. The curvature dependence of film tension can be compared with that of interfacial tension. Curvature dependence of surface tension depends on the locations of the Gibbs dividing surface and the surface of tension [16].

We can evaluate curvature dependence of film tension by using Eq. 6. Introducing Eq. 18 into Eq. 6, we have h

 . i 1−2 vAf þ vLA r dγ f ¼ −s f dT þ v f d p þ vL d pBL   .  −2 vAf þ vLA γ f r2 dr c X Γ if dμ i : − i¼1

ð64Þ Curvature dependence of film tension can then be expressed at constant chemical potentials by 

∂γ f =∂r

 T ; p; pBL ; μ

       ¼ −2 vAf þ vLA γ f =r2 = 1−2 vAf þ vLA =r ;

ð65Þ where μ stands for the set of μ1,…, μc. vAf +vAL is related to the distance hA between the surface of tension

Fig. 3 +f/+∞f vs. 1/r curves at constant hA =hA∞: (1) 2nm, (2) 5 nm. The number on the upper abscissa shows r (in nanometer) corresponding to 1/r

Colloid Polym Sci

Furthermore, curvature dependence of surface tension of a pure substance depends on the Gibbs adsorption at the surface of tension and the curvature dependence for a droplet is opposite to that for a bubble [17]. However, it is reasonable to assume rAF
pAB, n1f,⋯,ncf, the difference Δ μi in the chemical potential of component i between thin and thick films is expressed by

Interaction between interfaces in a film

Z Δμi ¼

As a film becomes thin, interaction occurs between film interfaces in the thin film and the film core is no longer homogeneous in the direction perpendicular to the film. The film tension and film interfacial tension of the thin film are different from those of the thick film. In this section, we develop the equations which allow one to evaluate the thermodynamic quantities of interaction between interfaces in a film from experiments, based on the thermodynamic formulation in the above sections. The excess Gibbs free energy of film follows from Eq. 44 Gf ¼

c X

nif μ i :

ð68Þ

i¼1

          ¼ μ i pBL ; γ f −μ i 0; γ f þ μ i 0; γ f −μ i 0; γ0f ; ð72aÞ where +f0 is the film tension of the thick film at pBL =0. Substitution of Eqs. 70 and 71 into Eq. 72a yields pBL 0

Z

BL vL;γ i dp −

γf γ0f

γ ai;0 dγ f ;

ð73Þ

+ being the ai+ at pBL =0. The mean molar excess Gibbs ai,0 free energy g f of film is defined by

gf ¼ Gf

c .X i¼1

nif ¼

c X

X if μ i ;

ð74Þ

i¼1

where the mole fraction Xif of component i in the film is defined by c .X . X if ¼ nif n fj ¼ Γ if Γ f ; ð75Þ total film density Г f being

dG f ¼ −S f d T þ V f dp þ V L d pBL   þ V Af þ V LA dpAB −σdγ f þ

Γf ¼ c X

μi dn if :

ð69Þ

The partial derivatives of μi with respect to pBL and +f at constant T, p, pAB, nf1,…,nfc can then be written as  T ; p; pAB ; γ f ; n f

  ¼ ∂V L =∂n if T ; p; pBL ; pAB ;γ f ; n f ¼ vL;γ i ; i

ð70Þ

  ∂μ i =∂γ f

T ; p; pBL ; pAB; n f

c X

Γ if :

ð76Þ

i¼1

i¼1

∂μ i =∂pBL

ð72Þ

j¼1

Combination of Eq. 6 with Eq. 68 gives



    Δμi ¼ μi pBL ; γ f −μi 0; γ0f

  ¼ − ∂σ=∂nif T ; p; pBL ; pAB ;γ f ; n f ¼ −aγi ; i

ð71Þ

Change Δg f in mean molar excess Gibbs free energy associated with interaction between film interfaces, which we call Gibbs free energy of interaction, can be defined by     Δg f ¼ g f pBL ; γ f −g f 0; γ0f ¼

c X

X if Δμ i ;

ð77Þ ð78Þ

i¼1

at constant T, p, pAB, X2f,…,Xcf, and Гf. Substituting Eq. 73 into Eq. 78 yields Z pBL Δg f ¼ v L d pBL −a f Δγ f ; ð79Þ 0

v iL,γ

where we defined the apparent molar volume and area a γi of component i in the film, taking into consideration that the numbers of moles of solvents are not kept constant in the partial derivatives. At a given T, p,

where we used the difference Δγf, in film tension Δγ f ¼ γ f −γ0f ;

ð80Þ

Colloid Polym Sci

the definition of mean molar volume vL and area a f , and their relations to the corresponding apparent molar quantities .X c c X vL ¼ VL nif ¼ X if vL;γ ð81Þ i ; i¼1

i¼1

c . X af ¼ 1 Γf ¼ X if aγi;0:

ð82Þ

Combination of Eq. 6 with Eq. 74 leads to

  f L dg f ¼ −s f dT þ v f dp þ v L dpBL þ vA þ vA dpAB −a f dγ f c X þ ðμi −μ1 ÞdX if i¼2

ð83Þ

i¼1

We can numerically evaluate Δg f by substituting the values of vL and a f obtained from the dependence of film tension on pBL and m1,…, mc into Eq. 79.

and then we have

      f L d Δg f ¼ −Δs f dT þ Δv f dp þ v L dpBL þ ΔvA þ ΔvA dpAB −a f d Δγ f c X ðΔμi −Δμ1 ÞdX if : þ

ð84Þ

i¼2

Here, mean molar quantity y f , thermodynamic quantities Δy f of interaction, and ΔvAf þ ΔvLA are defined by yf ¼ Yf

c .X

nif ; ðy ¼ s; vÞ

ð85Þ

i¼1

and     Δy f ¼ y f pBL ; γ f −y f 0; γ0f ;

ð86Þ

 K  K K ΔvA ¼ vA pBL ; γ f −vA 0; γ0f ; ðK ¼ f ; LÞ;

ð87Þ

respectively. From Eq. 84, entropy Δs f and volume Δv f of interaction and ΔvAf þ ΔvLA can be expressed by h   i Δs f ¼ − ∂ Δg f =∂T ð88Þ f BL AB f p; p ; p

Eqs. 88–92. Thermodynamic quantities of interaction per unit film area can be simply obtained by multiplying the corresponding thermodynamic quantities of interaction by the total film density. By putting pAB =0 and replacing pBL with the disjoining pressure Π of equilibrium plane-parallel films in the above formulation, we can obtain the corresponding equations for interaction between interfaces in a plane-parallel film: Z Π Δg f ¼ v L d Π−a f Δγ f ; ð93Þ 0

    d Δg f ¼ −Δs f dT þ Δv f dp þ v L dΠ−a f d Δγ f

; Δγ ; X ;

þ h   i Δv f ¼ ∂ Δg f =∂p

ðΔμ i −Δμ1 ÞdX if ;

ð94Þ

i¼2 T ; pBL ; pAB ; Δγ f ; X f ;

ð89Þ



f f L ΔvA þ ΔvA ¼ ∂ Δg =∂pAB

and h   i Δs f ¼ − ∂ Δg f =∂T

and

p;Π;Δγ f ;X f ;

ð95Þ

ð90Þ T ; p; pBL ; Δγ f ; X f ;

respectively. The corresponding energy Δu f and enthalpy f Δh are derived from Eqs. 58, 59, and 86:

  f L Δu f ¼ Δg f þ TΔs f −pΔv f −pBL v L −pAB ΔvA þ ΔvA þ Δγ f a f ;

ð91Þ Δh f ¼ Δg f þ TΔs f :

c X

h   i Δv f ¼ ∂ Δg f =∂p

T ;Π; Δγ f ;X f ;

ð96Þ

Δu f ¼ Δg f þ TΔs f −pΔv f −Π v L þ Δγ f a f ;

ð97Þ

Δh f ¼ Δg f þ TΔs f :

ð98Þ

ð92Þ

We can hence evaluate all the thermodynamic quantities of interaction from the Gibbs free energy of interaction using

for Eqs. 79, 84, 88, 89, 91, and 92, respectively.

Colloid Polym Sci

Application to simple cases of a four components system Let us consider spherical films in a four-component system (solvents a, b, l, and solute 1) as simple cases. Case in which film thickness is extremely small compared with film radius We can make the following assumptions for the spherical film whose thickness is extremely small compared with its radius, v h ¼ ht ;

ð99Þ

vhAt ¼ hA t ;

ð100Þ

Γ 1f ¼ Γ 1F;A þ Γ 1F;B ;

ð101Þ

Δy f ¼ ΔyAF þ ΔyBF ;

ð102Þ

ðy ¼ s; vÞ;

The above equations except Eq. 106 and Eq. 108 except the term of pAB are similar in form to the corresponding ones for plane-parallel films when pBL is replaced by disjoining pressure [4]. Case in which the phase L is an ideal solution If the phase L is assumed to be an ideal dilute solution, we have   L μ1 ¼ μm þ RT ln m1 ; 1 T; p

ð109Þ

where μ 1m is a standard chemical potential and μ11 ¼ RT =m1 ;

μ 1a ¼ μ1b ¼ 0:

Substituting Eq. 110 into Eqs. 48–52b yields   Δs f ¼ − ∂ γ f =∂T p; pBL ; pAB ; m 1

ð110Þ

ð111Þ

h   i  ¼ − 1−2 vAf ;A þ vh tA =r ∂ γ f =∂T p; pBL ; r; m 1

ð111aÞ

  ¼ −r−2 r2A ∂ γAF =∂T þ r 2B ∂ γBF =∂T −pBL ∂vh =∂T ;

ð111bÞ

Equations 48b, 49b, 50b, 51b, 52b, and 62a then reduce to 

 F

Δs f ¼ −∂ γ AF þ γ B =∂T −pBL ∂ht =∂T    þ D−1 Γ 1F;A þ Γ 1F;B Dbs μ1a þ Das μ1b ;   Δv f ¼ ∂ γ AF þ γ BF =∂p þ pBL ∂ht =∂p    þ D−1 Γ 1F;A þ Γ 1F;B Dbv μ1a þ Dav μ1b ; 

 F



  AB BL AB vAf ;A ¼ vAF;A ¼ ∂ γAF þ γBF =∂p  þ p ∂ht =∂p þD−1 Γ 1F;A þ Γ 1F;B DbA μ1a þ DaA μ1b ;

ð105Þ

ð106Þ

 F

Γ 1f μ11 ¼ −∂ γAFþ γB =∂m1 −pBL ∂ht =∂m1   þ D−1 Γ 1F;A þ Γ 1F;B Db1 μ1a þ Da1 μ1b ;

  Δv f ¼ ∂ γ f =∂p T ; pBL ; pAB ; m 1

ð112Þ

h   i  ¼ 1−2 vAf ;A þ vhAt =r ∂ γ f =∂p T ; pBL ; r; m 1

ð112aÞ

  ¼ r−2 r2A ∂ γAF =∂p þ r2B ∂ γBF =∂p þ pBL ∂vh =∂p;

ð112bÞ

ð104Þ



BL BL BL h f ¼ ∂ γAF þ  γB =∂p þ∂ p ht =∂p  F;A F;B þD−1 Γ 1 þ Γ 1 DbL μ1a þ DaL μ1b ;



ð103Þ

ð107Þ

  Δu f ¼ TΔs f −pΔv f −pBL h f −ht −pAB vAF;A þ γAF þ γBF : ð108Þ

  h f ¼ ∂ γ f =∂pBL T; p; pAB ; m 1

ð113Þ

h   i  ¼ 1−2 vAf ;A þ vhAt =r ∂ γ f =∂pBL T; p; r; m 1

ð113aÞ

  BL ¼ r−2 r2A ∂ γAF =∂p þ r2B ∂ γBF =∂pBL   þ ∂ pBL vh =∂pBL ;

ð113bÞ

  vAf þ vLA ¼ ∂ γ f =∂pAB T ; p; pBL ; m 1

ð114Þ

Colloid Polym Sci

      ¼ ð1=2Þr2 ∂ γ f =∂r = r ∂ γ f =∂r −γ f

ð114aÞ

  2 r−2 r2A vAf ;A ¼ r−2 r2A ∂ γAF =∂pAB þ rB ∂ γBF =∂pAB þ pBL ∂vh =∂pAB ;

Eqs. 45 and 47 reduce to       I I I I B I Δy f ¼ Γ Ia ya−yA a þ Γ b yb −yb þ Γ l yl −yl þ

c X

  f Γ if yi −yi ;

ð120Þ

i¼1

ð114bÞ and

  Γ 1f ¼ −ðm1 =RT Þ ∂ γ f =∂m 1 T ; p; pBL ; pAB

L

ð115Þ

     ¼ −ðm1 =RT Þ 1−2 vAf þ vLA =r ∂ γ f =∂m 1 T ; p; pBL ; r ð115aÞ     ¼ −ðm1 =RTÞ r−2 r2A ∂ γ AF =∂m 1 þ r2B ∂ γ BF =∂m1 þ pBL ∂vh =∂m 1 :

h f ¼ Γ Il =cl þ Γ 1f v1 ;

ð121Þ

respectively. The right-hand side of Eq. 120 is the change in thermodynamic quantity upon the formation of film interfaces from the given phases; hence, we call the Δyf thermodynamic quantity of film interface formation. Δuf and Δhf are given by the same equations as Eqs. 62, 62a, and 63, respectively.

ð115bÞ Δuf and Δhf are given by Eqs. 62, 62a, and 63, respectively. Case in which solvents are immiscible If the solvents are practically immiscible with one another, the concentration of solvent components in the phases A, B, and L are assumed to be zero except caA, cbB, and clL, and Eq. 13 then reduces to dμ 1 ¼ −s1 dT þ v1 dp−v1 dpBL þ μ11 dm 1

ð116Þ

for component 1. Substituting Eq. 116 into Eqs. 6 and 23, we obtain the same equations as Eqs. 111–114b for the case of the ideal solution, and Eqs. 52, 52a, and 52b reduce to   Γ 1f μ11 ¼ − ∂γ f =∂m 1 T; p; pBL ; pAB

ð117Þ

     ¼ − 1−2 vAf þ vLA =r ∂ γ f =∂m 1 T ; p; pBL ; r

Appendix 1 Let us consider the hypothetical system is mechanically equivalent to the actual system. The hypothetical system consists of three phases A, B, and L separated by two spherical surfaces of tension for the film interfaces concentric with the film. We define spherical coordinates (r, θ, 8) with the origin at the center of the curvature of the film and the polar axis directed outward through the center (Fig. 1b). The phase A is the portion between r=0 and rA and L is the one between rA and rB, and the phase B is between rB and ∞. Then, the work δW done on the system by a small strain is given by    δW ¼ −∭ p N ðrÞerr þ pT ðrÞ eθθ þ eφφ r2 sinθ d r d θd φ   ¼ −∭pABL ðrÞ err þ eθθ þ eφφ r2 sinθ d r d θdφ

Z     þ pABL ðrÞ−pT ðrÞ r2 d r∬ eθθ þ eφφ sinθ d θ dφ Z ¼− p

ð117aÞ

  ¼ −r−2 r2A ∂ γAF =∂m 1 þ r2B ∂ γBF =∂m 1 −pBL ∂vh =∂m 1 : ð117bÞ

∞ 0

Z   r2 d r þ pA −p

rA 0

Z   r2 d r þ pL −p

rB

r2 d r

rA

  ∬ eθθ þ eφφ sinθ dθdφ Z ∞  ABL    p ðrÞ−pT ðrÞ r2 d r∬ eθθ þ eφφ sinθ d θ d φ þ 0

  ¼ −pδV −pAB δ V A þ V AF;A þ pBL vh δσ þ γAF δσA þ γBF δσB ;

Since I;B I I Γ I;A a ¼ Γ a; Γ b ¼ Γ b;

ð118Þ

I Γ I;L l ¼ Γ l;

and thereby I

Δya ¼ ya−yA a;

I

I

Δyb ¼ yb−yBb ; Δyl ¼ yl −yl ;

ð119Þ

ð122Þ where pN (r) and pT (r) are the normal and tangential components of the pressure tensor at r, respectively, and err, eθθ, and eφφ are the diagonal components of the strain tensor [19,

Colloid Polym Sci

20], and from the condition of the mechanical equilibrium, pN (r) can be expressed by

The enthalpy H and Gibbs free energy G are defined by     H ¼ U þ pA V A þ V AF;A þ p V B þ V BF;B

pN ðrÞ ¼ pABL ðrÞ

  þ pL V L þ V AF;L þ V BF;L −γAF σA −γBF σB

¼ ½1−Aðr−rA ÞpA þ ½Aðr−rA Þ−Aðr−rB ÞpL þ Aðr−rB Þp;

  ¼ U þ pV þ pAB V A þ V AF;A −pBL vh σ−γAF σA −γBF σB

ð123Þ where

ð128Þ

and AðrÞ ¼ 1; ¼ 0;

ð r ≥ 0Þ ðr < 0Þ:

G ¼ H−TS˙: Therefore,

Here, we used the relations Z r  ABL 0  2 2 F r A γA ¼ p ðr Þ−pT ðr0 Þ ðr0 Þ d r0 ;

ð124Þ

  dG ¼ −S dT þ V d p þ V A þ V AF;A d pAB −σvh d pBL −σpBL dvh

0

Z r2B γBF ¼

ð129Þ

−σA d γAF −σB d γBF þ μa d na þ μb d nb þ μl d nl þ

c X

μ i d ni :

i¼1

∞

 2 pABL ðr0 Þ−pT ðr0 Þ ðr0 Þ d r0 ;

ð130Þ

ð125Þ

From Euler’s theorem,

r

G ¼ μa na þ μb nb þ μl nl þ

and

c X

μi ni :

ð131Þ

i¼1

  δω ¼ ∬ eθθ þ eφφ sinθ d θdφ ¼ δσ=r2 ¼ δσA =r2A ¼ δσB =r2B :

Combination of Eqs. 130 and 131 yields Eq. 19.

ð126Þ For the internal energy U of the system, we have from Eq. 122   dU ¼ TdS−pdV −p d V A þ V AF;A þ pBL vh dσ þ γAF dσA c X þ γBF dσB þ μa d na þ μ b d n b þ μ l d n l þ μi d ni : AB

i¼1

ð127Þ

Appendix 2 Substituting Eq. 60 at hA =hA∞ into Eq. 59 yields

Z  f f ln γ =γ∞ ¼

lnr ∞

i h  2   h∞A =r h∞A =r −3 h∞A =r þ 3 dlnr  ∞ 3  ∞ 2   hA =r −3 hA =r þ 3 h∞A =r −3=2 

  h  i   ¼ − ð1=3Þ 1−21=3 þ 22=3 ln 1− h∞A =r = 1 þ 2−1=3   

  h i2 2   ∞ −1=3 −2=3 −4=3 −8=3 −4=3 −8=3 þ2 þ32 þ32 −ð1=3Þ 1−2 = 1−2 ln hA =r − 1−2 þ

 n h  i h  io   3−1=2 21=3 þ 22=3 Tan−1 3−1=2 24=3 h∞A =r þ 3−1=2 1−24=3 −Tan−1 3−1=2 1−24=3 :

In the calculation of film tension, hA∞ was assumed to be one half of film thickness.

Appendix 3: Nomenclature ai + af ci K

ð132Þ

Apparent molar area of component i in film Mean molar area of film Number of moles of component i per unit volume in phase K (K=A, B)

ci (r′) G Gf gf Δg f Δgf H

Number of moles of component i per unit volume at r′ Gibbs free energy Excess Gibbs free energy of film Mean molar excess Gibbs free energy of film Gibbs free energy of interaction between film interfaces Gibbs free energy change associated with adsorption at film interfaces Enthalpy

Colloid Polym Sci

Δhf ht h At hA hA∞ hf mi ni  nK i pA p pL pBL pAB r rK rKK′ S si sK sf sKF Δsf T U Δuf V VK* VK vi viL,+ vL vf vKF Δvf F,L vF,K K ,vK f,A vA

vL

Enthalpy change associated with adsorption at film interfaces Distance between surfaces of tension for film interfaces Distance between surfaces of tension for film and inner film interface Distance between surface of tension for film and innermost dividing interface hA of plane-parallel film Thermodynamic film thickness Molality of component i Number of moles of component i Number of moles of component i in portion K* (K=A, B) Pressure in phase A Pressure in phase B Pressure in phase L Difference in pressure between phases B and L Difference in pressure between phases A and B Radius of film Radius of surface of tension for film interface against phase K (K=A, B) Radius of dividing surface (K=A, B;K′=F, L) Entropy Partial molar entropy of component i Entropy per unit volume in phase K (K=A, B, and L) Excess entropy per unit film area Excess entropies per unit area ascribed to film interface against phase K (K=A, B) Entropy change associated with adsorption at film interfaces Temperature Energy Energy change associated with adsorption at film interfaces Volume Volume of portion K* Volume of phase K (K=A, B, and L) Partial molar volume of component i Apparent molar volume of component i in film Mean molar volume of phase L Excess volume per unit film area Excess volumes per unit area ascribed to film interface against phase K (K=A, B) Volume change associated with adsorption at film interfaces Parts of vKF divided by surface of tension Volume (per unit film area) between innermost surface of tension and innermost dividing surface Volume of phase L per unit film area

vKL vh vhAt Δvf Xif yif yIi Δy f +f +0f +∞f +KF Γ fi Γf Γ iF,K Γ iI Γ iI,K μi μij Π σ σK ω

Parts of vL divided by surface of tension for film Volume (per unit film area) between surfaces of tension for film interfaces Volume (per unit film area) between surfaces of tension for film and for inner film interface Volume change associated with adsorption at film interfaces Mole fraction of component i in film Mean partial molar quantity of component i in film Mean partial molar quantity of component i inherent in film Thermodynamic quantity of interaction (y=s, v, u, h, and g) Film tension Film tension at pBL =0 Film tension of plane-parallel film Film interfacial tension against phase K (K=A, B) Film density of component i Total film density Interfacial density of component i ascribed to film interface against phase K (K=A, B) Film density of component i inherent in film Film density of component i ascribed to portion K of film region (K=A, B, and L) Chemical potential of component i Partial derivative of chemical potential of component i with respect to the molality of component j Disjoining pressure of plane-parallel film Film area Area of film interface against phase K (K=A, B) Solid angle

References 1. Israelachvili JN (1985) Intermolecular and surface forces, 2nd edn. Academic, London, Chap 18 2. Exerowa D, Kruglyakov PM (1997) Foam and foam films, volume 5: theory, experiment, application. In: Möbius D, Miller R (eds) Studies in interface science. Elsevier, Amsterdam, pp 282–303 3. de Feijter JA, Vrij A (1978) J Colloid Interface Sci 64:269 4. Iyota H, Krustev R, Müller H-J (2004) Colloid Polym Sci 282:1329 5. Iyota H, Krustev R, Müller H-J (2004) Colloid Polym Sci 282:1392 6. Iyota H, Ikeda N, Krastev R (2011) J Colloid Interface Sci 364:170 7. Ivanov IB, Kralchevsky PA (1988) Mechanics and thermodynamics of curved thin films. In: Ivanov IB (ed) Thin liquid films, surfactant science series 29. Dekker, New York, Chap 2 8. Motomura K, Iwanaga S, Hayami Y, Uryu S, Matuura R (1981) J Colloid Interface Sci 80:32 9. Motomura K, Iyota H, Aratono M, Yamanaka M, Matuura R (1983) J Colloid Interface Sci 93:264

Colloid Polym Sci 10. Prigogine I, Bellemans A (1966) In: Everett DH (trans.) Surface tension and adsorption. Longmans, London, Chap 6 11. Eriksson JC, Toshev BV (1982) Colloids Surf 5:241 12. Kralchevsky PA, Ivanov IB (1986) In: Mittal KL (ed) Surfactant in solution, vol 6. Plenum, New York, pp 1549–1556 13. Motomura K (1978) J Colloid Interface Sci 64:348 14. Ono S, Kondo S (1960) Handbuch der Physik, vol. 10. Springer, Berlin, pp 163–168 15. de Feijter JA (1988) Thermodynamics of thin liquid films. In: Ivanov IB (ed) Thin liquid films, surfactant science series 29. Dekker, New York, Chap 1

16. Guggenheim EA (1977) Thermodynamics, 6th edn. NorthHolland, Amsterdam, pp 55–58 17. Prigogine I, Bellemans A (1966) In: Everett DH (trans.) Surface tension and adsorption. Longmans, London, pp 256– 258 18. Motomura K, Baret JF (1983) J Colloid Interface Sci 91:391 19. Ono S, Kondo S (1960) Handbuch der Physik, vol. 10. Springer, Berlin, pp 157–163 20. Goodrich FC (1969) The thermodynamics of fluid interfaces. In: Matijevic E (ed) Surface and colloid science, vol 1. Wiley, New York